An exponent tells you how many times to multiply a base by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The rules below all follow from that one idea, which is useful when you forget one. Write out the multiplication and the rule appears.
The rules at a glance
| Rule | In symbols | Example |
|---|---|---|
| Product | am · an = am+n | x³ · x⁵ = x⁸ |
| Quotient | am / an = am−n | x⁷ / x² = x⁵ |
| Power of a power | (am)n = amn | (x³)⁴ = x¹² |
| Power of a product | (ab)n = anbn | (2x)³ = 8x³ |
| Power of a quotient | (a/b)n = an/bn | (x/3)² = x²/9 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 5⁰ = 1 |
| Negative exponent | a−n = 1/an | 2−3 = 1/8 |
| Fractional exponent | am/n = (ⁿ√a)m | 82/3 = 4 |
Product and quotient rules
When you multiply powers with the same base, add the exponents. x³ · x⁵ is three x’s times five x’s, which is eight x’s.
When you divide powers with the same base, subtract the exponents. In x⁷/x², two x’s on the bottom cancel two on the top, leaving x⁵.
Both rules need the same base. x³ · y⁵ can’t be combined, and neither can 2³ · 3⁵.
Power rules
When you raise a power to a power, multiply the exponents: (x³)⁴ = x³ · x³ · x³ · x³ = x¹².
An exponent outside parentheses applies to every factor inside:
Example Simplify (2x³)⁴
- Apply the exponent to each factor: 2⁴ · (x³)⁴.
- 2⁴ = 16 and (x³)⁴ = x¹².
- The answer is 16x¹².
Zero and negative exponents
Follow the quotient rule and you get these for free. x³/x³ = x3−3 = x⁰, and anything divided by itself is 1. So x⁰ = 1.
Similarly, x²/x⁵ = x−3. Written out, two x’s on top cancel with two of the five on the bottom, leaving 1/x³. So x−3 = 1/x³.
A negative exponent means “reciprocal.” It does not make the number negative: 2−3 = 1/8, which is positive.
Example Simplify (2/3)−2
Flip the fraction and make the exponent positive: (3/2)² = 9/4.
Fractional exponents
The denominator of a fractional exponent is a root and the numerator is a power. x1/2 = √x, and x1/3 = ∛x.
Example Evaluate 82/3
- Take the cube root first: ∛8 = 2.
- Then square: 2² = 4.
Taking the root first keeps the numbers small. 8² = 64 and ∛64 = 4 gives the same answer with more work.
Example Evaluate 27−1/3
- The negative means reciprocal: 1/271/3.
- ∛27 = 3, so the answer is 1/3.
Putting the rules together
Most problems mix several rules. Work through one at a time: clear parentheses with the power rules, then combine like bases, then rewrite negative exponents.
Example Simplify (3x²y)³ / (9xy²)
- Expand the top: 3³x⁶y³ = 27x⁶y³.
- Divide coefficients: 27/9 = 3.
- Subtract exponents for each base: x6−1 = x⁵ and y3−2 = y.
- The answer is 3x⁵y.
Example Simplify (x−2y³)−2
- Multiply each exponent by −2: x⁴y−6.
- Rewrite the negative exponent: x⁴/y⁶.
Scientific notation
Scientific notation writes a number as a value between 1 and 10 times a power of 10. The exponent rules make it quick to multiply and divide.
Example Multiply (3 × 10⁴)(2 × 10−7)
Multiply the front numbers and add the exponents: 6 × 10−3, which is 0.006.
Example Divide (8 × 10⁶) / (4 × 10²)
Divide the front numbers and subtract the exponents: 2 × 10⁴.
If the front number ends up 10 or more, shift it: 30 × 10⁸ = 3 × 10⁹.
Mistakes that cost points
Exponents don’t distribute over addition
(x + y)² is not x² + y². It is (x + y)(x + y) = x² + 2xy + y². Try x = 1, y = 1: (1 + 1)² = 4, while 1² + 1² = 2.
Watch the negative sign
−3² = −9, because the exponent applies only to the 3. (−3)² = 9, because the exponent applies to −3.
Add exponents when multiplying
x² · x³ = x⁵. Multiplying the exponents to get x⁶ is the rule for (x²)³, a different problem.
Know what the zero exponent touches
2x⁰ = 2 · 1 = 2. Only the x is raised to zero. (2x)⁰ = 1.
Practice problems
Simplify (x⁴)³ · x².
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x¹⁴. The power rule gives x¹², then the product rule adds 2.
Simplify (4a²b)² / (8ab³).
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2a³/b. The top is 16a⁴b². Dividing gives 2a³b−1.
Evaluate 163/4.
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8. The fourth root of 16 is 2, and 2³ = 8.
Evaluate (2/3)−2.
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9/4. Flip the fraction and square it.
Simplify x1/2 · x1/3.
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x5/6. Add the exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Write (5 × 10³)(6 × 10⁵) in scientific notation.
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3 × 10⁹. The product is 30 × 10⁸, and 30 = 3 × 10¹.
Simplify (x−3y²)² · x⁵.
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y⁴/x. Squaring gives x−6y⁴. Multiplying by x⁵ gives x−1y⁴.
Practice
Questions students ask
Why does anything to the zero power equal 1?
Each step down in the exponent divides by the base: 2³ = 8, 2² = 4, 2¹ = 2. One more step divides 2 by 2, which is 1. That pattern is why 2⁰ = 1.
What is 0⁰?
It is left undefined in most algebra courses. You won’t see it on standard tests.
Do these rules work for variables in the exponent?
Yes. 2x · 2³ = 2x+3. That is the starting point for solving exponential equations, covered in Logarithms.